Lexicographical polytopes (cid:73)

Within a fixed integer box of R n , lexicographical polytopes are the convex hulls of the integer points that are lexicographically between two given integer points. We provide their descriptions by means of linear inequalities. Throughout, (cid:96), u, r, s will denote integer points satisfying (cid:96) ≤ r ≤ u and (cid:96) ≤ s ≤ u , that is r and s are within [ (cid:96), u ]. A point x ∈ Z n is lexicographically smaller than y ∈ Z n , denoted by x (cid:52) y , if x = y or the first nonzero coordinate of y − x is positive. We write x ≺ y if x (cid:52) y and x (cid:54) = y . The lexicographical polytope P r (cid:52) s (cid:96),u is the convex hull of the integer points within [ (cid:96), u ] that are lexicographically between r and s : The top-lexicographical polytope is special case when . Similarly, the bottom-lexicographical polytope